Fast Hartley Transform Algorithms Research Articles

New Decimation-In-Time Fast Hartley Transform Algorithm

This paper presents a new algorithm for fast calculation of the discrete Hartley transform (DHT) based on decimation-in-time (DIT) approach. The proposed radix-2^2 fast Hartley transform (FHT) DIT algorithm has a regular butterfly structure that provides flexibility of different powers-of-two transform lengths, substantially reducing the arithmetic complexity with simple bit reversing for ordering the output sequence. The algorithm is developed through the three-dimensional linear index map and by integrating two stages of the signal flow graph together into a single butterfly. The algorithm is implemented and its computational complexity has been analysed and compared with the existing FHT algorithms, showing that it is significantly reduce the structural complexity with a better indexing scheme that is suitable for efficient implementation.

Improved FHT Algorithms for Fast Computation of the Discrete Hartley Transform

In this paper, by using the symmetrical properties of the discrete Hartley transform (DHT), an improved radix-2 fast Hartley transform (FHT) algorithm with arithmetic complexity comparable to that of the real-valued fast Fourier transform (RFFT) is developed. It has a simple and regular butterfly structure and possesses the in-place computation property. Furthermore, using the same principles, the development can be extended to more efficient radix-based FHT algorithms. An example for the improved radix-4 FHT algorithm is given to show the validity of the presented method. The arithmetic complexity for the new algorithms are computed and then compared with the existing FHT algorithms. The results of these comparisons have shown that the developed algorithms reduce the number of multiplications and additions considerably.

Area-efficient and scalable solution to real-data fast fourier transform via regularised fast Hartley transform

A highly parallel hardware solution to the regularised fast Hartley transform (FHT), an algorithm developed only recently for the efficient computation of the discrete Hartley transform (DHT) and hence of the real-data fast Fourier transform (FFT), is discussed. A drawback of conventional FHT algorithms lies in the loss of regularity arising from the need for two types of 'butterfly' for catering for fixed-radix formulations. A generic radix-4 double butterfly was therefore developed for the regularised FHT which overcame the problem in an elegant fashion. An architecture for the parallel computation of the generic double butterfly and the resulting regularised FHT is now discussed in some detail; this exploits a single high-performance processing element (PE) which yields an attractive solution, particularly when implemented with field-programmable gate array (FPGA) technology, which is both area-efficient and scalable in terms of transform length. High performance is achieved by having the PE able to process the input/ output data sets to the double butterfly in parallel, this in turn implying the need to be able to access simultaneously, and without conflict, both multiple data and multiple twiddle factors, or coefficients, from their respective memories. The resulting area-efficient and scalable single-PE architecture is shown to yield a generic solution to the real-data FFT which is capable of achieving the throughput of the most advanced commercially available solutions for just a fraction of the silicon area and, for new applications, at negligible re-design cost.

A Split Vector-Radix Algorithm for the 3-D Discrete Hartley Transform

In this paper, we propose a three-dimensional (3-D) split vector-radix fast Hartley transform (FHT) algorithm. The main idea behind the proposed algorithm is that the radix-2/4 approach is introduced in the decomposition of the 3-D discrete Hartley transform by using an appropriate index mapping and the Kronecker product. This provides an algorithm based on a mixture of radix-(2times2times2) and radix-(4times4times4) index maps and has a butterfly that is characterized by simple closed-form expressions. This algorithm offers substantial reductions in the numbers of multiplications, additions, data transfers, and twiddle factor evaluations or accesses to the look-up table, without a significant increase in the structural complexity compared to that of the existing 3-D vector radix FHT algorithm

Multidimensional vector radix FHT algorithms

In this paper, efficient multidimensional (M-D) vector radix (VR) decimation-in-frequency and decimation-in-time fast Hartley transform (FHT) algorithms are derived for computing the discrete Hartley transform (DHT) of any dimension using an appropriate index mapping and the Kronecker product. The proposed algorithms are more effective and highly suitable for hardware and software implementations compared to all existing M-D FHT algorithms that are derived for the computation of the DHT of any dimension. The butterflies of the proposed algorithms are based on simple closed-form expressions that allow easy implementations of these algorithms for any dimension. In addition, the proposed algorithms possess properties such as high regularity, simplicity and in-place computation that are highly desirable for software and hardware implementations, especially for the M-D applications. A close relationship between the M-D VR complex-valued fast Fourier transform algorithms and the proposed M-D VR FHT algorithms is established. This type of relationship is of great significance for software and hardware implementations of the algorithms, since it is shown that because of this relationship and the fact that the DHT is an alternative to the discrete Fourier transform (DFT) for real data, a single module with a little or no modification can be used to carry out the forward and inverse M-D DFTs for real- or complex-valued data and M-D DHTs. Thus, the same module (with a little or no modification) can be used to cover all domains of applications that involve the DFTs or DHTs.

Design and parallel computation of regularised fast Hartley transform

The paper describes the design and parallel computation of a regularised fast Hartley transform (FHT), to be used for computation of the discrete Fourier transform (DFT) of real-valued data. For the processing of such data, the FHT has attractions over the fast Fourier transform (FFT) in terms of reduced arithmetic operation counts and reduced memory requirement, whilst its bilateral property means it may be straightforwardly applied to both forward and inverse DFTs. A drawback, however, of conventional FHT algorithms lies in the loss of regularity arising from the need for two sizes of ‘butterfly’ for efficient fixed-radix implementations. A generic double butterfly is therefore developed for the radix-4 FHT which overcomes the problem in an elegant fashion. The result is a recursive single-butterfly solution, referred to as the regularised FHT, which lends itself naturally to parallelisation and to mapping onto a regular computational structure for implementation with algorithmically specialised hardware.

A New Split-Radix FHT Algorithm for Length-<tex>$q*2^m$</tex>DHTs

In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length N=q*2/sup m/, where q is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- q*2/sup m/ DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case.

Fast Computation of Circular Convolution of Real Valued Data using Prime Factor Fast Hartley Transform Algorithm

The paper presents both arithmetic and time complexities involved in the implementation of circular convolution of real valued data using various fast Hartley transform (FHT) algorithms. It is observed that the prime factor FHT (PFFHT) algorithm involves less arithmetic complexity and requires significantly less computation time compared with the radix-2 and split radix FHT algorithms for the implementation of circular convolution. Apart from that, the PFFHT offers closer choices for convolution lengths unlike the radix-2 or split radix algorithms.

Efficient fast Hartley transform algorithms for hypercube-connected multicomputers

Although fast Hartley transform (FHT) provides efficient spectral analysis of real discrete signals, the literature that addresses the parallelization of FHT is extremely rare. FHT is a real transformation and does not necessitate any complex arithmetics. On the other hand, FHT algorithm has an irregular computational structure which makes efficient parallelization harder. In this paper, we propose an efficient restructuring for the sequential FHT algorithm which brings regularity and symmetry to the computational structure of the FHT. Then, we propose an efficient parallel FHT algorithm for medium-to-coarse grain hypercube multicomputers by introducing a dynamic mapping scheme for the restructured FHT. The proposed parallel algorithm achieves perfect load-balance, minimizes both the number and volume of concurrent communications, allows only nearest-neighbor communications and achieves in-place computation and communication. The proposed algorithm is implemented on a 32 node iPSC/2 hypercube multicomputer, high-efficiency values are obtained even for small size FHT problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fixed-point error analysis of radix-4 FHT algorithm with optimised scaling schemes

The fixed-point error performance of the various fast Hartley transform (FHT) algorithms have been investigated. Scaling schemes have been proposed for each of the algorithms. However, due to their better error performance, only the decimation-in-time (DIT) FHT algorithms have been examined. The fixed-point error analysis of the radix-4 DIT algorithm is discussed first and is shown to agree closely with the simulation results. These results are then compared with the simulation results for radix-2 and split-radix algorithms. The scaling schemes are then optimised and the simulation results of the three algorithms are compared. It is concluded that the radix-4 DIT algorithm has the best error performance.

Fast radix-3/9 discrete Hartley transform

An efficient algorithm for computing radix-3/9 discrete Hartley transforms (DHTs) is presented. It is shown that the radix-3/9 fast Hartley transform (FHT) algorithm reduces the number of multiplications required by a radix-3 FHT algorithm for nearly 50%. For the computation of real-valued discrete Fourier transforms (DFTs) with sequence lengths that are powers of 3, it is shown that the radix-3/9 FHT algorithm reduces the number of multiplications by 16.2% over the fastest real-valued radix-3/9 fast Fourier transform (FFT) algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

New radix-3 and −6 decimation-in-frequency fast Hartley transform algorithms

Fast algorithms of a transform, like fast Fourier transform (FFT) algorithms, are based on different decomposition techniques. It is shown that these decomposition techniques can also be applied to the computation of the discrete Hartley transform (DHT) for a real-valued sequence. Recently, an efficient decomposition technique for radix-3 decimation-in-time (DIT) FFT and fast Hartley transform (FHT) algorithms has been demonstrated. Such a decomposition technique is implemented for radix-3 and -6 decimation-in-frequency (DIF) FHT algorithms and found to improve the operation count. Efficiency in these algorithms is derived by pairing the rotating factors with an appropriate reordering of the input sequence. From the results, it is seen that the radix-3 and -6 FHT algorithms presented are comparable to the split-radix FHT algorithm in terms of their operation count and will be more efficient when the sequence length is closer to an integer power of the corresponding radix.

Split vector-radix fast Fourier transform

The split-radix approach for computing the discrete Fourier transform (DFT) is extended for the vector-radix fast Fourier transform (FFT) to two and higher dimensions. It is obtained by further splitting the (N/2*N/2) transforms with twiddle factors in the radix (2*2) FFT algorithm. The generalization of this split vector-radix FFT algorithm to higher radices and higher dimensions is also presented. By introducing a general approach for constructing the fast Hartley transform (FHT) from the corresponding FFT, new vector- and split-vector-radix FHT algorithms with the same desirable properties as their FFT counterparts are obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fast Hartley transform pruning

The discrete Hartley transform (DHT) is discussed as a tool for the processing of real signals. Fast Hartley transform (FHT) algorithms which compute the DHT in a time proportional to N log/sub 2/ N exist. In many applications, such as interpolation and convolution of signals, a significant number of zeros are padded to the nonzero valued samples before the transform is computed. It is shown that for such situations, significant savings in the number of additions and multiplications can be obtained by pruning the FHT algorithm. The modifications in the FHT algorithm as a result of pruning are developed and implemented in an FHT subroutine. The amount of savings in the operation is determined. >

Computing the Fourier transform of real data on a hypercube

There are two ways, other than the standard fast Fourier transform (FFT) algorithm, of computing Fourier transforms of real data, namely, (1)the real fast Fourier transform (RFFT) algorithm, and (2) the fast Hartley transform (FHT) algorithm. On a sequential computer, it has been shown that both the RFFT and the FHT algorithms are faster than the FFT algorithm. However, it is not obvious that the same is true on a parallel machine. The communication requirements of the RFFT and the FHT algorithms, which are critical to the cost of any parallel implementation, are different from those of the FFT algorithm. In this paper we present efficient implementations of the RFFT and the FHT algorithms on a hypercube machine. Experimental results are given for the implementation of the RFFT and the FHT algorithms on the NCUBE machine.

Recursive Relations for Complexities of Fast Hartley Transform Algorithms

Recursive relations for the number of non-trivial summations and multiplications required for the various fast Hartley transform algorithms are derived. These relations show that Bracewell's decimation-in-time [3] and Meckelburg and Lipka's decimation-in-frequency [5] algorithms are equally fast and slower than the split radix algorithm due to Pei and Wu[1] and Sorensen et al [2].

Application of the fast Hartley transform to industrial noise analysis

The application of the fast Hartley transform (FHT) to industrial noise analysis to replace the fast Fourier transform (FFT) is proposed. The Hartley transform of a real‐valued function is also real‐valued. Its evaluation does not involve complex functions and it has the same formula for forward and inverse transform. Thus it has a speed advantage over FFT. The type of FHT algorithm chosen is due to H. V. Sorensen et al. [IEEE Trans. Acoust. Speech Signal Process. ASSP‐33, 1231–1238 (1985)]. As an illustration, the two‐channel FFT signal processor is chosen. The computation of the following functions is repeated: autocorrelation function, cross‐correlation function, power spectrum, and the acoustic intensity of real‐numbered data by replacing the FFT with FHT. A detailed analysis of the time that can be saved in computation by using FHT is made. A key advantage of the Hartley transform technique for large autocorrelation is the use of the same formula for transforming and inverse transforming.

Computation of discrete Hilbert transform through fast Hartley transform

A fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT). Instead of the conventional fast Fourier transform (FFT) approach, the processing is carried out entirely in the real domain. Also, since many efficient FHT algorithms exist, the computation complexity can be greatly reduced from two complex FFTs into two real FHTs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Fast Hartley Transform Algorithm

The fast Hartley transform (FHT) is similar to the Cooley-Tukey fast Fourier transform (FFT) but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the FFT. Through use of the FHT, discrete cosine transforms (DCT) and discrete Fourier transforms (DFT) can be obtained. The recursive nature of the FHT algorithm derived in this paper enables us to generate the next higher order FHT from two identical lower order FHT's. In practice, this recursive relationship offers flexibility in programming different sizes of transforms, while the orderly structure of its signal flow-graphs indicates an ease of implementation in VLSI.

Structured fast Hartley transform algorithms

The flowgraph originally proposed for the computation of the radix-2 fast Hartley transform (FHT) is restructured for clarity and ease of implementation with hardware and software. By applying the transposition theorem to the newly formed flowgraph (which corresponds to a decimation-in-time algorithm), a decimation-in-frequency algorithm with a similar flowgraph is obtained. The number of operations for the two algorithms is the same because of an existing symmetry, which is also identical to that of another FHT algorithm recently proposed.